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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 线性回归的从零开始实现\n",
    "\n",
    "在了解了线性回归的背景知识之后，现在我们可以动手实现它了。尽管强大的深度学习框架可以减少大量重复性工作，但若过于依赖它提供的便利，会导致我们很难深入理解深度学习是如何工作的。因此，本节将介绍如何只利用`torch`和`autograd`来实现一个线性回归的训练。\n",
    "\n",
    "首先，导入本节中实验所需的包或模块，其中的matplotlib包可用于作图，且设置成嵌入显示。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from IPython import display\n",
    "from matplotlib import pyplot as plt\n",
    "import torch\n",
    "from torch import autograd\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 生成数据集\n",
    "\n",
    "我们构造一个简单的人工训练数据集，它可以使我们能够直观比较学到的参数和真实的模型参数的区别。设训练数据集样本数为1000，输入个数（特征数）为2。给定随机生成的批量样本特征$\\boldsymbol{X} \\in \\mathbb{R}^{1000 \\times 2}$，我们使用线性回归模型真实权重$\\boldsymbol{w} = [2, -3.4]^\\top$和偏差$b = 4.2$，以及一个随机噪声项$\\epsilon$来生成标签\n",
    "\n",
    "$$\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{w} + b + \\epsilon,$$\n",
    "\n",
    "其中噪声项$\\epsilon$服从均值为0、标准差为0.01的正态分布。噪声代表了数据集中无意义的干扰。下面，让我们生成数据集。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "num_inputs = 2\n",
    "num_examples = 1000\n",
    "true_w = [2, -3.4]\n",
    "true_b = 4.2\n",
    "features = torch.randn(num_examples, num_inputs)\n",
    "labels = true_w[0] * features[:, 0] + true_w[1] * features[:, 1] + true_b\n",
    "labels += torch.normal(mean=torch.zeros(labels.shape), std=0.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "注意，`features`的每一行是一个长度为2的向量，而`labels`的每一行是一个长度为1的向量（标量）。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(tensor([-0.1986,  0.4762]), tensor(2.1801))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "features[0], labels[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "通过生成第二个特征`features[:, 1]`和标签 `labels` 的散点图，可以更直观地观察两者间的线性关系。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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   "source": [
    "def use_svg_display():\n",
    "    # 用矢量图显示\n",
    "    display.set_matplotlib_formats('svg')\n",
    "\n",
    "def set_figsize(figsize=(3.5, 2.5)):\n",
    "    use_svg_display()\n",
    "    # 设置图的尺寸\n",
    "    plt.rcParams['figure.figsize'] = figsize\n",
    "\n",
    "set_figsize()\n",
    "plt.scatter(features[:, 1].numpy(), labels.numpy(), 1); # 加分号只显示图"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们将上面的`plt`作图函数以及`use_svg_display`函数和`set_figsize`函数定义在`d2ltorch`包里。以后在作图时，我们将直接调用`d2ltorch.plt`。由于`plt`在`d2ltorch`包中是一个全局变量，我们在作图前只需要调用`d2ltorch.set_figsize()`即可打印矢量图并设置图的尺寸。\n",
    "\n",
    "\n",
    "## 读取数据\n",
    "\n",
    "在训练模型的时候，我们需要遍历数据集并不断读取小批量数据样本。这里我们定义一个函数：它每次返回`batch_size`（批量大小）个随机样本的特征和标签。"
   ]
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   "execution_count": 5,
   "metadata": {},
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   "source": [
    "# 本函数已保存在d2ltorch包中方便以后使用\n",
    "def data_iter(batc_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    random.shuffle(indices) # 样本的读取顺序是随机的\n",
    "    for i in range(0, num_examples, batc_size):\n",
    "        j = indices[i: min(i + batc_size, num_examples)]\n",
    "        yield features[j], labels[j] # 根据索引返回相应元素"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "让我们读取第一个小批量数据样本并打印。每个批量的特征形状为(10, 2)，分别对应批量大小和输入个数；标签形状为批量大小。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[-1.2317, -1.1869],\n",
      "        [-0.4982, -1.9512],\n",
      "        [-0.8829, -1.2815],\n",
      "        [-1.1189,  0.1515],\n",
      "        [-0.3336,  0.2809],\n",
      "        [ 0.5786, -0.4517],\n",
      "        [ 0.1778, -0.7508],\n",
      "        [ 1.8876,  0.6642],\n",
      "        [ 1.7146, -0.3624],\n",
      "        [-1.6180,  1.6439]])\n",
      "tensor([ 5.7940,  9.8359,  6.8010,  1.4328,  2.5826,  6.9058,  7.1057,  5.7095,\n",
      "         8.8571, -4.5947])\n"
     ]
    }
   ],
   "source": [
    "batch_size = 10\n",
    "\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    print(X)\n",
    "    print(y)\n",
    "    break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 初始化模型参数\n",
    "\n",
    "我们将权重初始化成均值为0、标准差为0.01的正态随机数，偏差则初始化成0。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "w = torch.normal(mean=torch.zeros(num_inputs, 1), std=0.01)\n",
    "b = torch.zeros(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "之后的模型训练中，需要对这些参数求梯度来迭代参数的值，因此我们需要创建它们的梯度。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0.], requires_grad=True)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w.requires_grad_()\n",
    "b.requires_grad_()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义模型\n",
    "\n",
    "下面是线性回归的矢量计算表达式的实现。我们使用`mm`函数做矩阵乘法。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linreg(X, w, b):  # 本函数已保存在d2ltorch包中方便以后使用\n",
    "    return torch.mm(X, w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义损失函数\n",
    "\n",
    "我们使用上一节描述的平方损失来定义线性回归的损失函数。在实现中，我们需要把真实值`y`变形成预测值`y_hat`的形状。以下函数返回的结果也将和`y_hat`的形状相同。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def squared_loss(y_hat, y):  # 本函数已保存在d2ltorch包中方便以后使用\n",
    "    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 定义优化算法\n",
    "\n",
    "以下的`sgd`函数实现了上一节中介绍的小批量随机梯度下降算法。它通过不断迭代模型参数来优化损失函数。这里自动求梯度模块计算得来的梯度是一个批量样本的梯度和。我们将它除以批量大小来得到平均值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sgd(params, lr, batch_size): # 本函数已保存在d2ltorch包中方便以后使用\n",
    "    for param in params:\n",
    "        param.data.sub_(lr * param.grad.data / batch_size) # 不能直接对param进行计算\n",
    "        param.grad.data.zero_() # 清空梯度"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 训练模型\n",
    "\n",
    "在训练中，我们将多次迭代模型参数。在每次迭代中，我们根据当前读取的小批量数据样本（特征`X`和标签`y`），通过调用反向函数`backward`计算小批量随机梯度，并调用优化算法`sgd`迭代模型参数。由于我们之前设批量大小`batch_size`为10，每个小批量的损失`l`的形状为(10, 1)。回忆一下[“自动求梯度”](../chapter_prerequisite/autograd.ipynb)一节。由于变量`l`并不是一个标量，运行`l.backward()`将对`l`中元素求和得到新的变量，再求该变量有关模型参数的梯度。\n",
    "\n",
    "在一个迭代周期（epoch）中，我们将完整遍历一遍`data_iter`函数，并对训练数据集中所有样本都使用一次（假设样本数能够被批量大小整除）。这里的迭代周期个数`num_epochs`和学习率`lr`都是超参数，分别设3和0.03。在实践中，大多超参数都需要通过反复试错来不断调节。虽然迭代周期数设得越大模型可能越有效，但是训练时间可能过长。而有关学习率对模型的影响，我们会在后面“优化算法”一章中详细介绍。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.041036\n",
      "epoch 2, loss 0.000155\n",
      "epoch 3, loss 0.000046\n"
     ]
    }
   ],
   "source": [
    "lr = 0.03\n",
    "num_epochs = 3\n",
    "net = linreg\n",
    "loss = squared_loss\n",
    "\n",
    "for epoch in range(num_epochs):  # 训练模型一共需要num_epochs个迭代周期\n",
    "    # 在每一个迭代周期中，会使用训练数据集中所有样本一次（假设样本数据能够被批量大小整除）。X\n",
    "    # 和y分别是小批量样本的特征和标签\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        l = loss(net(X, w, b), y) # l是有关小批量X和y的损失\n",
    "        l.backward(torch.ones_like(l)) # 小批量的损失对模型参数求梯度\n",
    "        sgd([w, b], lr, batch_size)  # 使用小批量随机梯度下降迭代模型参数\n",
    "    with torch.no_grad():\n",
    "        train_l = loss(net(features, w, b), labels)\n",
    "        print('epoch %d, loss %f' % (epoch + 1, train_l.mean().numpy()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "训练完成后，我们可以比较学到的参数和用来生成训练集的真实参数。它们应该很接近。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2, -3.4]\n",
      "tensor([[ 1.9993],\n",
      "        [-3.3991]], requires_grad=True)\n"
     ]
    }
   ],
   "source": [
    "print(true_w)\n",
    "print(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.2\n",
      "tensor([4.1997], requires_grad=True)\n"
     ]
    }
   ],
   "source": [
    "print(true_b)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 小结\n",
    "\n",
    "* 可以看出，仅使用`torch`和`autograd`模块就可以很容易地实现一个模型。接下来，本书会在此基础上描述更多深度学习模型，并介绍怎样使用更简洁的代码（见下一节）来实现它们。\n",
    "\n",
    "\n",
    "## 练习\n",
    "\n",
    "* 为什么`squared_loss`函数中需要使用`reshape`函数？\n",
    "* 尝试使用不同的学习率，观察损失函数值的下降快慢。\n",
    "* 如果样本个数不能被批量大小整除，`data_iter`函数的行为会有什么变化？\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "## 扫码直达[讨论区](https://discuss.gluon.ai/t/topic/743)\n",
    "\n",
    "![](../img/qr_linear-regression-scratch.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 参考\n",
    "* 如果y和y_hat的形状不一致相减会有广播机制,导致结果形状改变\n",
    "* data_iter产生的最后一个批量会是剩余的样本,而不是批量大小个样本"
   ]
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